$\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}19&8\\10&23\end{bmatrix}$, $\begin{bmatrix}25&32\\12&19\end{bmatrix}$, $\begin{bmatrix}29&0\\12&1\end{bmatrix}$, $\begin{bmatrix}29&8\\30&27\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
40.192.1-40.ca.1.1, 40.192.1-40.ca.1.2, 40.192.1-40.ca.1.3, 40.192.1-40.ca.1.4, 40.192.1-40.ca.1.5, 40.192.1-40.ca.1.6, 40.192.1-40.ca.1.7, 40.192.1-40.ca.1.8, 80.192.1-40.ca.1.1, 80.192.1-40.ca.1.2, 80.192.1-40.ca.1.3, 80.192.1-40.ca.1.4, 120.192.1-40.ca.1.1, 120.192.1-40.ca.1.2, 120.192.1-40.ca.1.3, 120.192.1-40.ca.1.4, 120.192.1-40.ca.1.5, 120.192.1-40.ca.1.6, 120.192.1-40.ca.1.7, 120.192.1-40.ca.1.8, 240.192.1-40.ca.1.1, 240.192.1-40.ca.1.2, 240.192.1-40.ca.1.3, 240.192.1-40.ca.1.4, 280.192.1-40.ca.1.1, 280.192.1-40.ca.1.2, 280.192.1-40.ca.1.3, 280.192.1-40.ca.1.4, 280.192.1-40.ca.1.5, 280.192.1-40.ca.1.6, 280.192.1-40.ca.1.7, 280.192.1-40.ca.1.8 |
Cyclic 40-isogeny field degree: |
$6$ |
Cyclic 40-torsion field degree: |
$96$ |
Full 40-torsion field degree: |
$7680$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} + y^{2} + z^{2} $ |
| $=$ | $5 y^{2} - 5 z^{2} + w^{2}$ |
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^8}{5^4}\cdot\frac{(625z^{8}-250z^{6}w^{2}+125z^{4}w^{4}-20z^{2}w^{6}+w^{8})^{3}}{w^{4}z^{8}(5z^{2}-w^{2})^{4}(10z^{2}-w^{2})^{2}}$ |
Hi
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Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
This modular curve is minimally covered by the modular curves in the database listed below.